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Chapter 38: Problem 2

Suppose \(A\) is a real symmetric \(805 \times 805\) matrix with eigenvalues \(1.00,1.01,1.02, \ldots, 8.98,8.99,9.00\) and also \(10,12,16,24 .\) How many steps of the conjugate gradient iteration must you take to be sure of reducing the initial error \(\left\|e_{0}\right\|_{A}\) by a factor of \(10^{6} ?\)

Short Answer

Expert verified
36 steps

Step by step solution

01

Understanding the Problem

The goal is to find how many steps are needed in the conjugate gradient iteration to reduce the error by a factor of \(10^6\). This requires understanding the eigenvalues of the matrix \(A\) and how they affect the convergence of the iteration.
02

List Given Eigenvalues

The eigenvalues of the matrix \(A\) are listed as \(1.00, 1.01, 1.02, \ldots, 8.98, 8.99, 9.00, 10, 12, 16, 24\). The eigenvalues range from \(1.00\) to \(24.00\).
03

Compute the Condition Number

The condition number, \(\kappa\), is the ratio of the largest eigenvalue to the smallest eigenvalue. Here, the largest eigenvalue is \(24\), and the smallest is \(1.00\). So, \(\kappa = \frac{24}{1.00} = 24\).
04

Apply Conjugate Gradient Convergence Formula

The convergence of the conjugate gradient method is given by the formula: \[ \frac{\left\| e_k \right\|_A}{\left\| e_0 \right\|_A} \leq 2 \left( \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1} \right)^k. \] We want this ratio to be less than \(10^{-6}\). Thus, we need: \[ 2 \left( \frac{\sqrt{24}-1}{\sqrt{24}+1} \right)^k < 10^{-6}. \]
05

Solve for k

First, find the numerical value of \(\frac{\sqrt{24}-1}{\sqrt{24}+1}\). \[ \sqrt{24} = 4.899. \] So, \[ \frac{4.899-1}{4.899+1} = \frac{3.899}{5.899} = 0.661. \] We need \(2(0.661)^k < 10^{-6}\). Take the natural logarithm of both sides to solve for \(k\): \[ \ln(2(0.661)^k) < \ln(10^{-6}). \] Simplify: \[ \ln(2) + k \ln(0.661) < -13.82. \] Since \(\ln(0.661) = -0.4145\): \[ \ln(2) + k(-0.4145) < -13.82 \] \[ 0.693 - 0.4145k < -13.82 \] \[ -0.4145k < -14.513 \] Divide both sides by -0.4145: \[ k > 35.02 \] Thus, \(k\) must be at least 36 steps.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Eigenvalues
Eigenvalues are special numbers associated with a square matrix. They provide insights into the matrix's behavior, particularly in linear transformations. For a matrix like the one in the exercise, its eigenvalues range from 1.00 to 24. Eigenvalues are important because they influence the convergence rate of iterative methods like the conjugate gradient method. In this context, these values determine how quickly the method can approximate the solution.
Condition Number
The condition number, denoted as \(\kappa\), is a measure of how sensitive a matrix's solution is to changes in the input. Specifically, it is calculated as the ratio of the largest eigenvalue to the smallest eigenvalue of the matrix. In the exercise, the largest eigenvalue is 24 and the smallest is 1. This gives a condition number of 24. A lower condition number indicates a well-conditioned matrix, leading to faster convergence of numerical methods. Conversely, a high condition number can signal slow convergence or numerical instability.

Convergence Rate
The convergence rate in iterative methods like conjugate gradient iteration refers to how quickly the sequence of approximations approaches the exact solution. It is tied closely to the condition number. The formula \(\frac{\left\| e_k \right\|_A}{\left\| e_0 \right\|_A} \leq 2 \( \frac{\sqrt{\kappa}-1}{\sqrt{\kappa}+1} \)^k\) encapsulates this relationship. In simpler terms, lower condition numbers result in faster convergence. From the problem, with \(\kappa = 24\), the convergence rate depends on the term \(\frac{\sqrt{24}-1}{\sqrt{24}+1}\). This formula underpins how many steps are needed to reach the desired accuracy.
Error Reduction
Error reduction in the context of iterative methods like the conjugate gradient method refers to progressively reducing the difference between the current approximation and the true solution. In the problem, we aim to reduce the initial error by a factor of \10^6\. The error reduction formula \(2 \( \frac{\sqrt{24}-1}{\sqrt{24}+1} \)^k \) helps determine the minimum number of iterations needed. By solving this, we find that at least 36 iterations are necessary to achieve the desired reduction. This underscores how the error diminishes exponentially with each step, depending on the condition number.

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Most popular questions from this chapter

Let \(A\) be the \(100 \times 100\) tridiagonal symmetric matrix with \(1,2, \ldots, 100\) on the diagonal and 1 on the sub-and superdiagonals, and set \(b=(1,1, \ldots, 1)^{T}\) Write a program that takes 100 steps of the \(\mathrm{CG}\) and also the steepest descent iteration to approximately solve \(A x=b .\) Produce a plot with four curves on it: the computed residual norms \(\left\|r_{n}\right\|_{2}\) for \(\mathrm{CG}\), the actual residual norms \(\left\|b-A x_{n}\right\|_{2}\) for \(\mathrm{CG},\) the residual norms \(\left\|r_{n}\right\|_{2}\) for steepest descent, and the estimate \(2(\sqrt{\kappa}-1)^{n} /(\sqrt{\kappa}+1)^{n}\) of Theorem \(38.5 .\) Comment on your results.
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